Skip to main content
Springer Nature Link
Log in

Fréchet mean-based Grassmann discriminant analysis

  • Special Issue Paper
  • Published:
Multimedia Systems Aims and scope Submit manuscript

Abstract

Representing image sets and videos with Grassmann manifold has become popular due to its powerful capability to extract discriminative information in machine learning research. However, existing techniques operations on Grassmann manifold are usually suffering from the problem of computational expensive, thus the application range of Grassmann manifold is limited. In this paper, we propose the Fréchet mean-based Grassmann discriminant analysis (FMGDA) algorithm to implement the videos (or image sets) data dimensionality reduction and clustering task. The data dimensionality reduction algorithm proposed by us can not only be used to reduce Grassmann data from high-dimensional data to a relative low-dimensional data, but also to maximize between-class distance and minimize within-class distance simultaneously. Fréchet mean is used to characterize the clustering center of Grassmann manifold space. We further show that the learning problem can be expressed as a trace ratio problem which can be efficiently solved. We designed a detailed experimental scheme to test the performance of our proposed algorithm, and the tests were assessed on several benchmark data sets. The experimental results indicate that our approach leads to a significant improvement over state-of-the-art methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Absil, P.A., Mahony, R., Sepulchre, R.: Riemannian geometry of grassmann manifolds with a view on algorithmic computation. Acta Appl. Math. 80(2), 199–220 (2004)

    Article  MathSciNet  Google Scholar 

  2. Yongli Hu, Boyue Wang, J.G.Y.S.H.C.M.A.B.Y.: Locality preserving projections for grassmann manifold. In: Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, IJCAI-17, pp. 2893–2900 (2017). https://doi.org/10.24963/ijcai.2017/403

  3. Centingu, H.E., Vidal, R.: Intrinsic mean shift for clustering on Stiefel and Grassmann manifolds. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1896–1902. IEEE (2009)

  4. Chan, A.B., Vasconcelos, N.: Modeling, clustering, and segmenting video with mixtures of dynamic textures. IEEE Trans. Pattern Anal. Mach. Intell. 30(5), 909–926 (2008)

    Article  Google Scholar 

  5. Dalal, N., Triggs, B.: Histograms of oriented gradients for human detection. In: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR), pp. 886–893. IEEE (2005)

  6. Edelman, A., Arias, T.A., Smith, S.T.: The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20(2), 303–353 (1998)

    Article  MathSciNet  Google Scholar 

  7. Fei-Fei, L., Fergus, R., Perona, P.: Learning generative visual models from few training examples: an incremental bayesian approach tested on 101 object categories. Comput. Vis. Image Underst. 106(1), 59–70 (2007)

    Article  Google Scholar 

  8. Fletcher, P.T., Venkatasubramanian, S., Joshi, S.: The geometric median on riemannian manifolds with application to robust atlas estimation. NeuroImage 45(1), S143–S152 (2009)

    Article  Google Scholar 

  9. Hamm, J., Lee, D.D.: Grassmann discriminant analysis: a unifying view on subspace-based learning. In: Proceedings of the 25th International Conference on Machine Learning (ICML), pp. 376–383. ACM (2008)

  10. Hamm, J., Lee, D.D.: Extended grassmann kernels for subspace-based learning. In: Advances in neural information processing systems (NIPS), pp. 601–608. Curran Associates (2009)

  11. Harandi, M., Sanderson, C., Shen, C., Lovell, B.C.: Dictionary learning and sparse coding on grassmann manifolds: an extrinsic solution. In: Proceedings of the IEEE International Conference on Computer Vision (CVPR), pp. 3120–3127. IEEE (2013)

  12. Harandi, M.T., Sanderson, C., Shirazi, S., Lovell, B.C.: Graph embedding discriminant analysis on grassmannian manifolds for improved image set matching. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 2705–2712. IEEE (2011)

  13. Huang, Z., Wang, R., Shan, S., Chen, X.: Projection metric learning on grassmann manifold with application to video based face recognition. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 140–149. IEEE (2015)

  14. Karcher, H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30(5), 509–541 (1977)

    Article  MathSciNet  Google Scholar 

  15. Kim, T.K., Kittler, J., Cipolla, R.: Discriminative learning and recognition of image set classes using canonical correlations. IEEE Trans. Pattern Anal. Mach. Intell. 29(6), 1005–1018 (2007)

    Article  Google Scholar 

  16. Krizhevsky, A., Hinton, G.: Learning multiple layers of features from tiny images. Technical Report. University of Toronto. https://www.cs.toronto.edu/~kriz/learning-features-2009-TR.pdf (2009)

  17. Liu, G., Lin, Z., Yan, S., Sun, J., Yu, Y., Ma, Y.: Robust recovery of subspace structures by low-rank representation. IEEE Trans. Pattern Anal. Mach. Intell. 35(1), 171–184 (2013)

    Article  Google Scholar 

  18. Marrinan, T., Ross Beveridge, J., Draper, B., Kirby, M., Peterson, C.: Finding the subspace mean or median to fit your need. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1082–1089 (2014)

  19. Ngo, T.T., Bellalij, M., Saad, Y.: The trace ratio optimization problem for dimensionality reduction. SIAM J. Matrix Anal. Appl. 31(5), 2950–2971 (2010)

    Article  MathSciNet  Google Scholar 

  20. Nie, F., Xiang, S., Jia, Y., Zhang, C., Yan, S.: Trace ratio criterion for feature selection. AAAI 2, 671–676 (2008)

    Google Scholar 

  21. Nishiyama, M., Yamaguchi, O., Fukui, K.: Face recognition with the multiple constrained mutual subspace method. In: International Conference on Audio-and Video-Based Biometric Person Authentication, pp. 71–80. Springer, New York (2005)

    Google Scholar 

  22. Rodriguez, M.D., Ahmed, J., Shah, M.: Action mach a spatio-temporal maximum average correlation height filter for action recognition. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1–8. IEEE (2008)

  23. Soomro, K., Zamir, A.R.: Action recognition in realistic sports videos. In: Computer Vision in Sports, pp. 181–208. Springer, New York (2014)

    Chapter  Google Scholar 

  24. Srivastava, A., Klassen, E.: Bayesian and geometric subspace tracking. Adv. Appl. Probab. 36(1), 43–56 (2004)

    Article  MathSciNet  Google Scholar 

  25. Wang, B., Hu, Y., Gao, J., Sun, Y., Chen, H., Ali, M., Yin, B.: Locality preserving projections for grassmann manifold. In: Proceedings of the 26th International Joint Conference on Artificial Intelligence, pp. 2893–2900. AAAI Press, San Francisco (2017)

  26. Wang, B., Hu, Y., Gao, J., Sun, Y., Yin, B.: Low rank representation on grassmann manifolds. In: Asian Conference on Computer Vision, pp. 81–96. Springer, New York (2014)

    Chapter  Google Scholar 

  27. Wang, B., Hu, Y., Gao, J., Sun, Y., Yin, B.: Product grassmann manifold representation and its LRR models. In: Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, pp. 2122–2129. AAAI Press (2016)

  28. Wang, B., Hu, Y., Gao, J., Sun, Y., Yin, B.: Laplacian LRR on product grassmann manifolds for human activity clustering in multicamera video surveillance. IEEE Trans. Circ. Syst. Video Technol. 27(3), 554–566 (2017)

    Article  Google Scholar 

  29. Wang, H., Yan, S., Xu, D., Tang, X., Huang, T.: Trace ratio vs. ratio trace for dimensionality reduction. In: IEEE Conference on Computer Vision and Pattern Recognition, IEEE, pp. 1–8 (2007)

  30. Wang, Y., Mori, G.: Human action recognition by semilatent topic models. IEEE Trans. Pattern Anal. Mach. Intell. 31(10), 1762–1774 (2009)

    Article  Google Scholar 

  31. Wong, Y.C.: Differential geometry of grassmann manifolds. Proc. Natl. Acad. Sci. 57(3), 589–594 (1967)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The School of Digital Media, Jiangnan University, Wuxi, Jiangsu, China

    Hongbin Yu, Yizhang Jiang & Pengjiang Qian

  2. China University of Mining Technology, Xuzhou, Jiangsu, China

    Kaijian Xia

Authors
  1. Hongbin Yu
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Kaijian Xia
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Yizhang Jiang
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Pengjiang Qian
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Pengjiang Qian.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported in part by the National Natural Science Foundation of China under Grants 61772241 and 61702225, by the Natural Science Foundation of Jiangsu Province under Grant BK20160187, by the Fundamental Research Funds for the Central Universities under Grant JUSRP51614A, by 2016 Qinglan Project of Jiangsu Province, by 2016 Six Talent Peaks Project of Jiangsu Province, and by the Science and Technology Demonstration Project of Social Development of Wuxi under Grant WX18IVJN002.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yu, H., Xia, K., Jiang, Y. et al. Fréchet mean-based Grassmann discriminant analysis. Multimedia Systems 26, 63–73 (2020). https://doi.org/10.1007/s00530-019-00629-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00530-019-00629-5

Keywords